Physical Interpretation

This section provides a physical interpretation of equations 3.7, which will lead to a deeper understanding of the mechanics of inversion.

The causal Green's function g_c and the acausal Green's g_a function for a point source in a 3-D homogeneous medium are given by:

 

$$g_c(\bf {r},t\vert\bf {r}_s,t_s) \delta(\tau -\vert\bf {r}-\bf {r}_s\vert/c)/r_{rs},$$ =

$$g_a(\bf {r},t\vert\bf {r}_s,t_s) \delta(\tau+\vert\bf {r}-\bf {r}_s\vert/c)/r_{rs},$$ = (3.10)

where c is the medium velocity, t_s denotes the source initiation time, t denotes the listening time, and $\tau=t-t_s$. A causal Green's function is typically used for forward modeling, while the acausal (waves are propagating prior to the source time) Green's function is used for backward propagation.

Note that these Green's functions are time invariant, which means that their values depend on the time lag between the source time t_s and listening time t. This is useful, because it means that a CSG collected in May should look the same as one collected in July for the same recording geometry and experimental site. In general, the Green's function for acoustic fields are time-invariant in an arbitrary velocity model (Morse and Feshbach, 1953).

Figure 3.5 shows that these Green's functions describe either a backward or forward light cone, where the apex of the cone kisses the source point at the time t_s. For a buried point source each cone intersects the surface plane z=0 along an hyperbola. The causal Green's function emulates exploding wavefronts from a point "source", while the acausal Green's function emulates imploding wavefronts from a point "sink".

Taking the Fourier transform of equation 3.11 w/r to the $\tau$ variable says:

 

$$\tilde G_c(\bf {r},t\vert\bf {r}_s) e^{i\omega \vert\bf {r}-\bf {r}_g\vert/c}/r_{rs},$$ =

$$\tilde G_a(\bf {r},t\vert\bf {r}_s) e^{-i\omega \vert\bf {r}-\bf {r}_g\vert/c}/r_{rs}.$$ =

or, $\tilde G_c(\bf {r},t\vert\bf {r}_s) = \tilde G_a(\bf {r},t\vert\bf {r}_s)^*$. This relation between the acausal and causal Green's functions is valid for arbitrary elastic media (Morse and Feshbach, 1953).

  

Figure 3.5: Forward and backward light cones due to a buried point source.